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We present statistical convergence results for the learning of (possibly) non-linear mappings in infinite-dimensional spaces. Specifically, given a map $G_0:\mathcal X\to\mathcal Y$ between two separable Hilbert spaces, we analyze the problem of recovering $G_0$ from $n\in\mathbb N$ noisy input-output pairs $(x_i, y_i)_{i=1}^n$ with $y_i = G_0 (x_i)+\varepsilon_i$; here the $x_i\in\mathcal X$ represent randomly drawn 'design' points, and the $\varepsilon_i$ are assumed to be either i.i.d. white noise processes or subgaussian random variables in $\mathcal{Y}$. We provide general convergence results for least-squares-type empirical risk minimizers over compact regression classes $\mathbf G\subseteq L^\infty(X,Y)$, in terms of their approximation properties and metric entropy bounds, which are derived using empirical process techniques. This generalizes classical results from finite-dimensional nonparametric regression to an infinite-dimensional setting. As a concrete application, we study an encoder-decoder based neural operator architecture termed FrameNet. Assuming $G_0$ to be holomorphic, we prove algebraic (in the sample size $n$) convergence rates in this setting, thereby overcoming the curse of dimensionality. To illustrate the wide applicability, as a prototypical example we discuss the learning of the non-linear solution operator to a parametric elliptic partial differential equation.

We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity, we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of popular statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors, in which both the proposed optimisation and sampling schemes provide good numerical recovery. The reproducible code is available online at https://github.com/MattGiord/LF-Diffusion., Comment: 50 pages, 9 figures, 5 tables, to appear in the Annals of Statistics

We exhibit examples of high-dimensional unimodal posterior distributions arising in non-linear regression models with Gaussian process priors for which MCMC methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialised (`cold start') algorithms that are local in the sense that their step-sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as pCN and MALA., Comment: 27 pages, 5 figures, to appear in Philosophical Transactions of the Royal Society A

The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevin-type MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data. Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local `geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schr\"odinger equation., Comment: 70 pages; to appear in the Journal of the European Mathematical Society

We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$, and the unique solution $u_f$ of the boundary value problem \[L_fu=g_1\text{ on } \mathcal O, \quad u=g_2 \text{ on }\partial \mathcal O,\] is observed corrupted by additive Gaussian white noise. Here $\mathcal O$ is a bounded domain in $\mathbb R^d$ with smooth boundary $\partial \mathcal O$, and $g_1, g_2$ are given functions defined on $\mathcal O, \partial \mathcal O$, respectively. Concrete examples include $L_fu=\Delta u-2fu$ (Schr\"odinger equation with attenuation potential $f$) and $L_fu=\text{div} (f\nabla u)$ (divergence form equation with conductivity $f$). In both cases, the parameter space \[\mathcal F=\{f\in H^\alpha(\mathcal O)| f > 0\}, ~\alpha>0, \] where $H^\alpha(\mathcal O)$ is the usual order $\alpha$ Sobolev space, induces a set of non-linearly constrained regression functions $\{u_f: f \in \mathcal F\}$. We study Tikhonov-type penalised least squares estimators $\hat f$ for $f$. The penalty functionals are of squared Sobolev-norm type and thus $\hat f$ can also be interpreted as a Bayesian `MAP'-estimator corresponding to some Gaussian process prior. We derive rates of convergence of $\hat f$ and of $u_{\hat f}$, to $f, u_f$, respectively. We prove that the rates obtained are minimax-optimal in prediction loss. Our bounds are derived from a general convergence rate result for non-linear inverse problems whose forward map satisfies a modulus of continuity condition, a result of independent interest that is applicable also to linear inverse problems, illustrated in an example with the Radon transform., Comment: 40 pages, to appear in SIAM/ASA Journal of Uncertainty Quantification